How to wrap an object around a cloth and "squeeze" the cloth, like a napkin ring or rag on a jar
For starters, I'd say model the thing manually if its not necessary to use the cloth sim.
Cloth sim is for interactivity, i.e. if you want the cloth to interact with something later in an animation.

However, if you want to use the cloth sim, try using Cloth Sewing (new since 2.70), here is a general tutorial: https://www.youtube.com/watch?v=ySyQsLD_7GM

In your case, i would use a similar approach as in the video with the cloak, wrap the cloth mesh around the jar lid, connect all open vertices with edges and mark them as seams to pull together.

I would model the string holding it together entirely manually.